What is the implicit derivative of #y=x^3y+x^2y^4-5y #?

1 Answer
Aug 24, 2016

#dy/dx =- (2x^2y + 3xy^2)/(x^3 + 2x^2y - 6)#

Explanation:

Let's first put all our variables on one side.

#x^3y + x^2y^2 - 5y - y = 0#

By a combination of implicit differentiation and the product rule, we can differentiate without having to isolate #y#.

#3x^2y + x^3(dy/dx) + 2xy^2 + 2yx^2(dy/dx) - 6(dy/dx) = 0#

#x^3(dy/dx) + 2x^2y(dy/dx) - 6(dy/dx) = -3x^2y - 2xy^2#

#dy/dx(x^3 + 2x^2y - 6) = -3x^2y - 2xy^2#

#dy/dx =- (2x^2y + 3xy^2)/(x^3 + 2x^2y - 6)#

Hopefully this helps!